Tuesday, February 12, 2008

Developmental Dyscalculia: A Brain-Based Etiology

Neurological Evidence for a Brain-Based Etiology of Dyscalculia

Anecdotal case-studies of patients with various brain lesions have demonstrated the dissociation of different calculation elements, thereby supporting the assumption that numerical ability represents a multifactor skill, requiring the participation of different abilities and quite diverse brain areas (Ardilla and Rosselli, 2002). These case studies have also allowed for the subtyping of dyscalculia and acalculia. Mathematical and arithmetic abilities can be impaired as a result of language, spatial, or executive functions. Ardilla and Rosselli (2002) detail some of the subtypes and associated ROIs (some of which are labeled on the diagram below):

Anarithmetia could be interpreted as a defect in understanding how the numerical system works, and is associated with damage to the left angular gyrus. Damage to the left angular gyrus is also associated with Gerstmann’s syndrome, which combines dyscalculia with finger agnosia (and results in an inability to count on one’s fingers), as well as dysgraphia and right-left disorientation. When electrical stimulation is applied to the angular gyrus in otherwise normal individuals, they present with signs of Gerstmann’s syndrome.

Patients with acalculia in Broca’s aphasia present with errors in the syntax of calculation. That is, they present “stack errors” (e.g. 14 is read as 4). While counting forward is not affected, counting backward relies more on verbal sequencing, and is impaired. Errors in transcoding numbers from verbal code to numerical code are present (e.g. “three hundred and seven” to 307), as are hierarchical errors (e.g. patients do not understand the difference between the two times the word “hundred” appears in “three hundred thousand, two hundred”). As this is associated with Broca’s aphasia, it is associated with the left inferior frontal gyrus.

Patients with acalculia in Wernicke’s aphasia present semantic and lexical errors in saying, reading, and writing numbers. However, simple mental arithmetic operations are errorless. Like in Broca’s aphasia, most of the errors that present in this case are language related. As these symptoms are associated with Wernicke’s aphasia, the left posterior superior temporal gyrus is implicated.

Patients with spatial acalculia have no difficulties in counting or in performing successive operations. However, some fragmentation appears in reading numbers (e.g. 523 becomes 23), resulting from left hemi-spatial neglect. Reading complex numbers is also prone to errors, as the spatial position of each digit relative to the other digits becomes important: 1003 becomes 103, 32 becomes 23, or 734 becomes 43. When writing, patients cannot line up numbers in columns, creating difficulty in arithmetic calculation. Moreover, digit iterations are frequent (e.g. 27 becomes 22277), as are feature iterations (e.g. 3 is written with extra loops). The patient has a full understanding of “carrying over” in subtraction, but cannot find the proper location to write the number.

Patients with frontal (executive function) acalculia have damage in the pre-frontal cortex. These patients typically present with serious difficulties in mental arithmetic operations, successive operations (particularly subtraction), and solving multi-step numerical problems. They generally also have serious disturbances in applying mathematical knowledge to time (e.g. they could not tell you if America was founded closer to 10 years ago or to 200 years ago). When aided by pencil and paper, however, most of these patients are errorless.

As the quality and quantity of different types of non-invasive neuroimaging methods has increased, researchers have been able to examine different regions of interest throughout the brain to discover how they are involved in mathematics and arithmetic, and how they can be implicated in developmental dyscalculia. Dehaene et al. (2004) carried out a series of fMRI investigations, in a study called Arithmetic and the Brain. They found a set of parietal, prefrontal, and cingulate areas which were reliably activated by patients undergoing mental calculation. The precentral sulcus is often co-activated with the inferior frontal gyrus. They’ve also considered the role of the left and right fusiform gyri and occipito-temporal regions in recognizing visual number forms.

Dehaene (2004) has implicated the angular gyrus in mathematics and arithmetic. The angular gyrus has been activated by digit naming tasks as well as mental multiplication. This was demonstrated by a study in which a normal patient’s angular gyrus was electrically stimulated, which disrupted multiplication. In addition, metabolic abnormalities have been found in the angular gyrus in individuals with dyscalculia: a focal defect in a left temporo-parietal brain region near the angular gyrus was isolated, with differential decreases in N-acetyl-aspartate, creatine, and choline (Levy, Reis, and Grafman, 1999).

A region of interest that has received lots of attention in dyscalculia research is the horizontal segment of the intraparietal sulcus (HIPS), in both hemispheres. Activation of the right and left HIPS has been seen during basic calculation tasks as well as digit detection tasks. Further, is it multi-modal, responding equally to spoken words and written words, as well as Arabic numerals. Right HIPS activation has also been seen in tasks where subjects estimate the numerosity of a set of concrete visual objects. Electrical stimulation of an anterior left HIPS site disrupted subtraction. Isaacs et al. (2001) found a left IPS reduction in grey matter in children with developmental dyscalculia at the precise coordinates where activation is observed in normal children during arithmetic tasks.

Molko et al. (2003) studied individuals with Turner Syndrome, a genetic X-linked condition which is associated with abnormal development of numerical representation. In the right IPS, a decrease in maximal depth as well as a trend toward reduced length was observed for subjects with Turner Syndrome when compared with control subjects. Additionally, the center of gravity of the central sulcus showed a significant posterior displacement in Turner Syndrome patients.

Despite the relative inter-subject irregularity of cortical geometry, there are general consistencies found in normal individuals. For example, the anterior-posterior orientation of the IPS, its downward convexity, as well as its segmentation into three parts, was observed in all controls. In contrast, the right intraparietal sulcal pattern of most subjects with Turner Syndrome did not conform to those patterns due to aberrant branches, abnormal interruption, or unusual orientation. For example, the three segments were only observed in 7 of 14 Turner Syndrome subjects, while the downward convexity was only seen in 3 of 14.

In agreement with the fMRI findings of Dehaene et al. (2004), during exact and approximate calculation tasks, Molko et al. (2003) found reduced activation in the right IPS as a function of number size. Similar fMRI hypoactivations were found in a broader parieto-prefrontal network in two other genetic conditions associated with developmental dyscalculia: fragile X syndrome and velocardiofacial syndrome (Dehaene et al., 2004).

In a meta-analysis of fMRI studies of arithmetic and numbers, Dehaene offer a tripartite organization for number processing in the brain:
The horizontal segment of the intraparietal sulcus (HIPS) appears as a plausible candidate for domain specificity: It is systematically activated whenever numbers are manipulated, independently of number notation, and with increasing activation as the task puts greater emphasis on quantity processing. Depending on task demands, we speculate that this core quantity system, analogous to an internal “number line,” can be supplemented by two other circuits. A left angular gyrus area, in connection with other left-hemispheric perisylvian areas, supports the manipulation of numbers in verbal form. Finally, a bilateral posterior superior parietal system supports attentional orientation on the mental number line, just like on any other spatial dimension. (Dehaene, Piazza, Pinel, and Cohen, 2003, p.1)

Sunday, February 10, 2008

Developmental Dyscalculia, Part Next

Continuing on in the series about a fascinating (I think) developmental learning disorder that not many people know about.

Cognitive Domains: Attention

Also associated with information processing theory is inhibition, which is the active suppression of irrelevant sensory input. Related to this is the idea of resistance to interference, or attention, which is the ability of an individual to concentrate on “central” information and ignore “peripheral” information. Normally-achieving students can complete an arithmetic worksheet in a noisy classroom with minimal distraction, and accuracy is usually very high. Students with developmental dyscalculia, however, may have issues with processing due to a deficit in inhibition.

Rosenberger (1989) offers evidence that low achievement in math is related to attentional deficits. He sampled 102 children for his study, and ran them on a series of paper-and-pencil tests and questionnaires; those children for whom the math achievement quotient was below 100, the reading achievement quotient above 100, and the difference between the two was at least 20 points (approximately 1.5 SDs) or greater were designated “dyscalculic.” Children who met the converse criteria were designated “dyslexic.” 72 children qualified as dyscalculic, and 30 qualified as dyslexic. Both groups were neurologically intact, and without history of epileptic seizures or structural central nervous system disease. The groups were highly comparable in overall scholastic aptitude as well; in fact, only the arithmetic score pre-experimentally distinguished the two groups.

Rosenberger found that the “freedom from distractibility” quotient from the Weschler scale was lower for the dyscalculics, although this is confounded with the score of the arithmetic subtest. Of four factors calculated from the DSM-III questionnaire that each participant received, only the factor of inattention was significantly different for the groups, and was higher for dyscalculics. Rosenberger offers that specific math underachievement is, in at least some cases, the result of failure of children with attention deficits to automatize number facts in the early grades. If true, “this finding would suggest that [attention deficit] is not merely an additive or aggravating factor in problems with math performance, but in fact interferes with the development of aptitude for this skill” (Rosenberger, 1989, p. 219).

Rosenberger, P.B. (1989). Perceptual-motor and attentional correlates of developmental dyscalculia. Annals of Neurology, 26, 216-220.

Wednesday, February 6, 2008

Developmental Dyscalculia, Part 4

Part 4 in the series:
Cognitive Domains: Memory

At the end of a child’s exploration of various strategies available, the mastery of elementary arithmetic is achieved when all basic facts can be retrieved from long-term memory without error. Mastery of basic arithmetic is crucial to later competence in more complex mathematical operations such as long division, fractions, geometry, calculus, and so on. Given the importance of memory to basic arithmetic competence, even if a child has successfully reached the most efficient strategy, deficits in memory could lead to mathematics disabilities.

When a computation is executed, the probability of direct retrieval increases for each subsequent solution to the same problem. However, in order for the execution of a computational strategy to lead to the construction of a long-term memory representation between a problem and its solution, both the equation’s augend (i.e. the first number) and addend (i.e. the second number), as well as the answer, must all be simultaneously active in working memory. Thus, arithmetic and mathematical ability is directly related to the function (or dysfunction) of the working and long-term memory stores (Geary, 1993).

In order to create a long-term memory for an arithmetic fact, such as 13 + 7 = 20, an individual must be both proficient (i.e. accurate) and efficient (i.e. speedy). Accuracy is important because if the child commits many computational errors, then the child is more likely to retrieve incorrect answers from long-term memory when later presented with the same problem. Efficiency is likewise important because with a slow counting speed, the working memory representation of the augend is more likely to decay before the addend and solution have been fully represented in working memory. In this circumstance, even if the child reaches the correct answer, it will be less strongly associated with the problem in long-term memory.

These cognitive models are evidenced empirically, as indicated by Geary (1993): “Cognitive studies indicate that when solving arithmetic problems, in relation to their normal peers, [mathematically disabled] children tend to use immature problem-solving strategies, have rather long solution times, and frequently commit computational and memory-retrieval errors.”

Butterworth (2005) further refines the role of working and long-term memory in the storage of arithmetic facts. He presents evidence that retrieval times show a very strong problem-size effect for single-digit problems: the larger the sum or product, the longer it takes to solve. Further, adults without any mathematical disability are quicker to solve an equation in the form of “larger addend” + “smaller addend” than they are to solve the same equation where the addends are reversed. Similarly, normal Italian children 6-10 years old took longer to solve a “smaller” x “larger” equation than a “larger” x “smaller” equation, despite the fact that the Italian education system teaches “smaller” x “larger” first (e.g. the 2x multiplication table is learned before the 6x multiplication table). This seems contradictory to the earlier theory, which offers that equations with which you have more experiences are more strongly stored in long-term memory – since the 2x arithmetic facts were presumably encoded into long-term memory well before the 6x arithmetic facts. This evidence suggests a more complex numerical organization to the storage and representation of arithmetic facts in long-term memory, not just rote association.

Information processing theory offers yet another model for the role that the function or dysfunction of working and long-term memory has in the pathology of developmental dyscalculia and other mathematical impairments. Central to information processing theory is the idea of limited capacity: the human mind has only a finite capacity for information processing at any one time. A fundamental assumption to this theory is that each type of mental process takes up some amount of the “space” or “energy”. At one extreme are automatic processes, which require virtually no space or energy. These processes work without intention or conscious awareness, don’t interfere with other processes, don’t improve with practice, and are not influenced by intelligence, education, motivation, or anything else, such as breathing or sweating. On the other end of the continuum are effortful processes, which use up the resources available in working memory, and have the opposite properties of automatic processes.

When confronted with an arithmetic task, a normal student can complete the task with minimal problems and fairly efficiently – even if the solution isn’t accessed via fact retrieval. For a student with math disabilities, however, the process is likely laborious and takes up significant amounts of energy. Perhaps this has something to do with the continuum of automatic and effortful processing. When a normally-achieving student is confronted with a straightforward arithmetic problem such as 5 + 11 + 37, the student can quickly identify the steps needed to solve the equation and move on to the next item on the worksheet. When a mathematically disabled student is confronted with the same problem, even after having learned and understood the fundamentals of counting and addition, each of the steps necessary to compute the answer takes up significantly more effort to complete. By the time the student moves on to the next item, he has already expended considerably more energy than the first student has, and has likely taken more time to complete each problem. After the first three or four equations, his energy store is perhaps depleted, and the rest of the worksheet is riddled with errors because the student has no mental energy left to tackle the subsequent calculations.